### 3D domain decomposition method coupling conforming and nonconforming finite elements

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This paper deals with the solution of problems involving partial differential equations in ${\mathbb{R}}^{3}$. For three dimensional case, methods are useful if they require neither domain boundary regularity nor regularity for the exact solution of the problem. A new domain decomposition method is therefore presented which uses low degree finite elements. The numerical approximation of the solution is easy, and optimal error bounds are obtained according to suitable norms.

We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ (${P}_{\lambda}$) ⎩ ${u}_{\mid \partial \Omega}=0$ where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem (${P}_{\lambda}$) admits a non-zero, non-negative strong solution ${u}_{\lambda}\in {\bigcap}_{p\ge 2}{W}^{2,p}\left(\Omega \right)$ such that $li{m}_{\lambda \to 0\u207a}\left|\right|{u}_{\lambda}{\left|\right|}_{{W}^{2,p}\left(\Omega \right)}=0$ for all p ≥ 2. Moreover, the function $\lambda \mapsto {I}_{\lambda}\left({u}_{\lambda}\right)$ is negative and decreasing in ]0,λ*[, where ${I}_{\lambda}$ is the energy functional related to (${P}_{\lambda}$).

Towards a constructive method to determine an L∞-conductivity from the corresponding Dirichlet to Neumann operator, we establish a Fredholm integral equation of the second kind at the boundary of a two dimensional body. We show that this equation depends directly on the measured data and has always a unique solution. This way the geometric optics solutions for the L∞-conductivity problem can be determined in a stable manner at the boundary and outside of the body.

In this paper, we employ the reduced basis method as a surrogate model for the solution of linear-quadratic optimal control problems governed by parametrized elliptic partial differential equations. We present a posteriori error estimation and dual procedures that provide rigorous bounds for the error in several quantities of interest: the optimal control, the cost functional, and general linear output functionals of the control, state, and adjoint variables. We show that, based on the assumption...

Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. We focus on mortar finite element methods on non-matching triangulations. In particular, we discuss and analyze dual Lagrange multiplier spaces for lowest order finite elements. These non standard Lagrange multiplier spaces yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces. As a consequence, standard efficient iterative...

We construct a bounded domain $\Omega \subset {\mathbb{R}}^{2}$ with the cone property and a harmonic function on Ω which belongs to ${W}_{0}^{1,p}\left(\Omega \right)$ for all 1 ≤ p < 4/3. As a corollary we deduce that there is no ${L}^{p}$-Hodge decomposition in ${L}^{p}(\Omega ,{\mathbb{R}}^{2})$ for all p > 4 and that the Dirichlet problem for the Laplace equation cannot be in general solved with the boundary data in ${W}^{1,p}\left(\Omega \right)$ for all p > 4.

This paper deals with the problem of finding positive solutions to the equation -∆[u] = g(x,u) on a bounded domain 'Omega' with Dirichlet boundary conditions. The function g can change sign and has asymptotically linear behaviour. The solutions are found using the Mountain Pass Theorem.